Two researchers have proved that Penrose tilings, famous patterns that never repeat, are mathematically equivalent to a kind of quantum error correction.
The most recommended introductory books on category theory (at least for pure mathematicians) are probably those by Steve Awodey, Tom Leinster, and Emily Riehl. All three have very considerable virtues.
This is a short, animated visual proof demonstrating two formulas for the sum of the first n positive cubes, one in terms of consecutive odd numbers and the other in terms of the square of a sum of consecutive positive integers. We end with a bonus visualization of a direct version of Nicomachus's theorem.
K. Ren, and H. Wang. (2023)cite arxiv:2308.08819Comment: 23 pages. v2: fixed small typo in abstract and added more details to arguments, main results unchanged.
M. Escardó. (2019)cite arxiv:1911.00580Comment: 211 pages, extended version of Midlands Graduate School course (2019), includes Agda-verified mathematics. Sources available at github (as explained in the pdf file), but not in LaTeX, last revised September 2022.
M. Porter. (2022)cite arxiv:2201.07794Comment: working paper (associated with my data-ethics lecture at the 2021 AMS Short Course on Mathematical and Computational Methods for Complex Social Systems); not yet refereed; private comments and suggestions are appreciated.
J. Llorca, and M. Delgado-Rodríguez. Medical science monitor : international medical journal of experimental and clinical research, 8 (10):
MT193-7(October 2002)3579<m:linebreak></m:linebreak>Risc atribuïble.